\section{Conclusion}
\label{Conclusion}
The report treats the bomber-battleship problem for two- and three-move lag and differentiates between a ship 
which has the possibility to stay still and not.
It also handles the construction of a discrete time, continuous state model for the problem. 
This conclusion summarizes the findings and results.

Ferguson's strategies for the bomber and the ship described in \cite{Ferguson} are $(\epsilon)$-optimal 
for the two-move lag. The simulated game between those two strategies gives the expected, theoretical value. 
%By trying different probabilities to move forwards $p$ against the optimal bomber the ship can improve by taking another probability than 
%the one worked out in \cite{Ferguson}.


For the two-move lag problem it is possible to find an optimal strategy for the ship taking the last step into account.
But the application of that idea does not work for the three-move lag problem and, using the equalizing idea of section \ref{Equalizing},  no optimal strategy for the ship taking the last steps into account could be found. 
Therefore Ferguson's strategy and the strategy from Lee and Lee are worked out for the three-move lag. Both strategies are 
not optimal for the ship but they try to reduce the hit rate. 
The bomber observes the ship and depending on its average movements 
the bomber chooses the place to bomb. Against \textit{ship Ferguson} the bomber works quite well, but the ship can improve by using the strategy
\textit{LeeLee}. Against this strategy the observing bomber does not achieve a hit rate up to the upper bound, hence it shows that the observing bomber is not optimal.

The comparison of the observing bomber against the optimal ship strategy for the two-move lag shows that it is not as good as the optimal strategy, but 
it comes close. It may improve by observing more moves before bombing.

If the ship can stay still in the two-move lag problem, Ferguson's strategy has to be adjusted, since there is the additional probability to stay still. 
With this option the ship can improve against the observing bomber. 
The bigger the size of the $n$-restricted graph, the less the improvement.

In continuous state model, the ship moves randomly through the plane, but there are some restrictions to simulate a realistic movement, 
and the bomber decides where to bomb based on its observations. It does not make a significant difference if 
the bomber takes the average or the weighted average of the ship's movement to predict where it will be when the bomb explodes. 
Against a ship which movements are defined by a function with increasing gradient, the weighted bomber is a bit better than the unweighted, because it values
the recent information (which is more important in this case) more.

It is possible to transform the continuous state model into a hidden Markov model with different states and emissions. In this case the ship moves according to 
the model, and the bomber tries to detect the model in order to determine the place to bomb.
The hidden Markov bomber needs some information about the model, otherwise the weighted and unweigthed bombers perform better. The hidden Markov bomber has to find the transion and emission probabilities and the current state of the ship. It bombs the 
place with the highest probability, according to the detected probabilities. 
If some of the information is given, the hit rate can be increased.


\paragraph{Future work}
For the three-move lag problem, the staying still problem and the continuous state model only some solutions were researched by this paper. There is definitely more interesting work to be done in these fields.
For instance, for the three-move lag problem an optimal ship strategy could be developed in the same spirit as Ferguson did for the two-move lag. Or for the hidden Markov model, the bomber uses standard Matlab functions, so if these functions could be adjusted specifically for our problem, then the bomber might improve. Moreover, there were ideas for using stochastic tracking principles for the bomber in the countinuous state model, but because of time constraints, this could not be pursued.
In short, this paper tried to explore some solutions for all of the problems, but it sacrificed some depth for more breadth in the research.
